1. No category

NAME
PERIOD
DATE
SIMPLE PENDULUM
Driving Question | Objective
What variables affect the period of a pendulum? Determine the physical properties of a simple
pendulum that affect its period.
Materials and Equipment
• Data collection system
• PASCO Smart
Gate1
• Meter stick
photogate
• Table clamp or large base
• PASCO Photogate Pendulum Set2
• Support rod, 60 cm or taller
• PASCO Pendulum Clamp3
• Thread
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• Scissors
1www.pasco.com/ap21
2www.pasco.com/ap10
3www.pasco.com/ap15
PASCO Smart Gate
PASCO Photogate
Pendulum Set
PASCO Pendulum
Clamp
Background
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How is a playground swing like a pendulum? Its motion is periodic; the swing's motion is repeated in
equal intervals of time. What produces this repetitive motion? The motion of both a swing and a
pendulum is the product of a variable force acting on them, called a linear restoring force. This force
is proportional to the horizontal distance x the swing is displaced from its equilibrium position.


F = −k x
(1)

 mg  
F = −
x

 l 
For a pendulum, m is the mass of the pendulum "bob," g is earth's gravitational acceleration
constant, and l is the length of the pendulum arm.
The equilibrium position for a pendulum is defined as the point at which the net force (and torque)
acting on the pendulum bob is zero. For the swing, the equilibrium position is the point at which the
force from gravity is counteracted completely by the tension from the swing's chains. This is where
the swing hangs motionless below its point of anchor. When a swing or pendulum is displaced from
its equilibrium position, the restoring force then acts on the swing, inducing motion back toward the
equilibrium position. For a pendulum, the speed of the bob increases as it approaches the
equilibrium position, at which point it achieves maximum speed.
Once the bob passes through the equilibrium position, its speed decrease as it moves up, away from
the equilibrium position, until it stops at the peak of its swing. At this point the bob begins to fall
back toward the equilibrium position, again subject to the same restoring force, and the changing
speed cycle starts over.
PASCO / PS-2848
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SIMPLE PENDULUM / STRUCTURED
Procedure
Part 1 – Displacement and Period
SET UP
1. Mount the pendulum clamp near the top of the rod and then cut a
length of thread about 10 cm longer than the distance between the
pendulum clamp and the top of the lab table.
2. Choose one of the four pendulum bobs from the PASCO Photogate
Pendulum Set (all have identical volume but different mass) and tie
one of the loose ends of the thread to the hook on the bob.
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3. Hang the pendulum from the third anchor point on the clamp
(farthest from the rod): loosen the anchor's thumbscrew and run the
thread under the anchor. Tighten the thumbscrew to hold the
thread in place.
4. Adjust the pendulum arm length (thread length) so the bob hangs
approximately 4 cm above the lab table.
5. With the pendulum bob hanging motionless, place the photogate on
the lab table with the arms of the photogate pointed upward,
directly under the pendulum bob. The pendulum bob should swing
freely between the arms on the photogate.
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6. Connect the photogate to your data collection system.
7. Configure the data collection system to measure the period of a pendulum using the photogate
and then create a table display with the period measurement as one of the columns.
COLLECT DATA
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8. Use your hand to pull the pendulum bob back, displacing it a
horizontal distance of 3 cm from its equilibrium position. Use the
meter stick to measure the horizontal displacement.
9. Begin recording data, and then release the pendulum bob so it swings
freely through the photogate.
10. Stop recording data when the data collection system has recorded 10
period measurements.
11. Use the tools on your data collection system to determine the average
of the 10 period data points. Record this average value as well as the
horizontal displacement into Table 1 in the Data Analysis section.
12. Repeat the same collect data steps 3 more times, increasing the horizontal displacement by an
additional 3 cm each trial. Record your average period and displacement values for each trial
into Table 1.
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PASCO / PS-2848
SIMPLE PENDULUM / STRUCTURED
Part 2 – Mass and Period
COLLECT DATA
13. Use the Part 1 setup for Part 2.
14. Remove the pendulum bob currently attached to the thread and then measure the individual
mass of all four pendulum bobs. Record the mass in order from smallest to largest in Table 2 in
the Data Analysis section.
15. Attach the pendulum bob with the smallest mass to the thread.
16. Use your hand to pull the pendulum bob back, displacing it a horizontal distance of 6 cm from its
equilibrium position. Use the meter stick to measure the horizontal displacement.
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17. Begin recording data, and then release the pendulum bob so it swings freely through the
photogate.
18. Stop recording data when the data collection system has recorded 10 period measurements.
19. Use the tools on your data collection system to determine the average of the 10 period data
points. Record this value next to its corresponding pendulum bob mass in Table 2.
20. Repeat the same data collection steps 3 more times, keeping the horizontal displacement
constant for each trial and changing the pendulum bob, using a bob with increasing mass each
time. Record the average period for each trial in Table 2.
COLLECT DATA
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Part 3 – Length and Period
21. Use the Part 1 setup for Part 3. Use any of the four pendulum bobs, but use the same pendulum
bob for each Part 3 trial.
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22. Use the meter stick to measure the length of the pendulum arm. Record this length in Table 3.
23. Use your hand to pull the pendulum bob back, displacing it a horizontal distance of 6 cm from its
equilibrium position. Use the meter stick to measure the horizontal displacement.
24. Begin recording data, and then release the pendulum bob so it swings freely through the
photogate.
25. Stop recording data when the data collection system has recorded 10 period measurements.
26. Use the tools on your data collection system to determine the average of the 10 period data
points. Record this value next to its corresponding pendulum arm length in Table 3.
27. Repeat the same collect data steps 4 more times, keeping the horizontal displacement constant
for each trial and shortening the length of the pendulum arm by 10 cm each time. Record your
average period for each trial into Table 3.
NOTE: To shorten the pendulum arm length, loosen the anchor thumbscrew on the pendulum
clamp and gently pull the loose end of the thread upward under the anchor. Tighten the anchor
thumbscrew again to hold the thread in place and then lower the pendulum clamp, with the
thread and bob attached, so that the bob hangs about 4 cm above the lab table in each trial.
PASCO / PS-2848
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SIMPLE PENDULUM / STRUCTURED
Data Analysis
Part 1 – Displacement and Period
Table 1: Period of a pendulum with varying horizontal displacement
Trial
Horizontal Displacement
(cm)
Average Period
(s)
1
2
3
4
1. Plot a graph of average period versus horizontal displacement in Graph 1. Be sure to label both
axes with the correct scale and units.
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Graph 1: Period versus displacement of a simple pendulum with constant length and mass
2. Did changing the displacement of the pendulum bob affect the period of the simple pendulum?
Justify your answer.
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PASCO / PS-2848
SIMPLE PENDULUM / STRUCTURED
Part 2 – Mass and Period
Table 2: Period of a pendulum with varying mass
Trial
Bob Mass
(g)
Average Period
(s)
1
2
3
4
3. Plot a graph of average period versus bob mass in Graph 2. Be sure to label both axes with the
correct scale and units.
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Graph 2: Period versus mass of a simple pendulum with constant length and displacement
4. Did changing the mass of the pendulum bob affect the period of the simple pendulum? Justify
your answer.
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SIMPLE PENDULUM / STRUCTURED
Part 3 – Arm Length and Period
Table 3: Period of a pendulum with varying arm length
Trial
Pendulum Arm Length
(cm)
Average Period
(s)
Pendulum Arm Length
(m1/2)
1
2
3
4
5
5. Plot a graph of average period versus pendulum arm length in Graph 3. Be sure to label both
axes with the correct scale and units.
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Graph 3: Period versus arm length of a simple pendulum with constant displacement and mass
6. Did changing the length of the pendulum arm affect the period of the simple pendulum? Justify
your answer.
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7. Calculate the square root of the pendulum arm length from Table 3. Record the result for each
trial in Table 3.
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PASCO / PS-2848
SIMPLE PENDULUM / STRUCTURED
8. Plot a graph of average period versus
axes with the correct scale and units.
arm length of a simple pendulum with constant displacement and mass
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e
Graph 4: Average period versus
pendulum arm length in Graph 4. Be sure to label both
9. Based on Graph 4, what is the relationship between period and pendulum arm length for a
simple pendulum (proportional, inverse, squared, et cetera)? Justify your answer.
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Analysis Questions
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1. For each part of your experiment, list each variable involved and state whether it was held
constant, increased, or decreased.
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2. In your experiment, what variables (physical properties) affected the period of a simple
pendulum and how did they affect the period?
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SIMPLE PENDULUM / STRUCTURED
3. The mathematical equation describing the period Tp of a pendulum is:
Tp = 2p
l
g
(2)
where l is the length of the pendulum arm and g is earth’s gravitational acceleration constant.
Does your data support this mathematical relationship? Justify your answer.
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Synthesis Questions
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1. In 1851, French physicist Leon Foucault created an enormous pendulum designed to
demonstrate the rotation of the earth. The pendulum consisted of a 28.0 kg bob hung from a
67.0 m cable in the Pantheon in Paris, France. Given this information, what is the period of
oscillation for the pendulum? Show your work.
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2. If the maximum displacement θ0 of Foucault's pendulum bob while in motion is 0.0149 rad, what
is the maximum linear speed of the bob during its motion? What is the maximum momentum of
the bob? Show your work.
3. One of two identical grandfather pendulum clocks from Earth is placed on Mars. If the
grandfather clock on Mars is 23 minutes behind the grandfather clock on Earth after 1 hour,
what is the acceleration due to gravity on Mars?
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PASCO / PS-2848
SIMPLE PENDULUM / STRUCTURED
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4. The figure to the right shows a person holding a 4 kg
pendulum bob hanging from a 6-m long cable. The cable is
attached to the side of an over-hanging wall 3 m up from the
bottom of the wall. If the person lets go of the bob, how long
will it take for it to swing back to him?
PASCO / PS-2848
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